Course in Probability Theory

8 1 DISTRIBUTION FUNCTIONb ithe size at jump at ai, thenF(ai) - F(ai-) = bisince F(ai +) = F(ai) .Consider the functionFd(x) =J(x)which represents the sum of all the jumps of F in the half-line (-00, x) . It is .clearly increasing, right continuous, with(3)Fd (-oc) = 0, Fd (+oo) =Hence Fd is a bounded increasing function . It should constitute the "jumpingpart" of F, and if it is subtracted out from F, the remainder should be positive,contain no more jumps, and so be continuous . These plausible statements willnow be proved -they are easy enough but not really trivial .iTheorem 1 .2 .1 .LetF, (x) = F(x) - FAX) ;then F, is positive, increasing, and continuous .PROOF .Let x < x', then we have(4) Fd(x') - FAX) =x